Analysis of Over-Dispersed Count Data: Application to Obligate
Parasite Pasteuria penetrans
IOANNIS VAGELAS
Department of Agriculture Crop Production and Rural Environment,
University of Thessaly, Volos,
GREECE
Abstract: - In this article we present with STATA regression models suitable for analyzing over-dispersed
count outcomes. Specifically, the Negative Binomial regression can be an appropriate choice for modeling
count variables, usually for over-dispersed count outcome variables. The common problem with count data
with zeroes is that the empirical data often show more zeroes than would be expected under either Poisson or
the Negative Binomial model. We concluded, this publications showcases that Zero-inflated models can be
used to model count data that has excessive zero counts.
Keywords: - Biological control; P. penetrans; over-dispersion; Excess zero-count data; Zero Inflation.
Received: April 25, 2021. Revised: January 5, 2022. Accepted: January 23, 2022. Published: March 1, 2022.
1 Introduction
Plant-parasitic nematodes such as the Meloidogyne
species are recognized as major agricultural
pathogens worldwide. Meloidogyne javanica is one
of the most damaging crop parasites often causing
heavy losses. Nowadays, the most reliable practices
to control the pathogen are preventive e.g. crop
rotation including the choice of plant varieties or the
use of biological control agents such as the obligate
hyperparasitic bacterium Pasteuria penetrans [1].
The literature review shows that the most widely
studied bacterial pathogen of Meloidogyne species
(root-knot nematodes) is in the genus Pasteuria.
Pasteuria penetrans is a mycelial, endospore-
forming, bacterial parasite that has shown
remarkable potential as a biological control agent of
second-stage juvenile (J2) of root-knot nematodes.
The biological control potential of Pasteuria spp.
has been demonstrated on many crops and has been
reported to develop endospores only in females of
Meloidogyne spp. [1]. Based on previous research
[2], attachment count data were observed to be over-
dispersed concerning high numbers of spores
attaching on each J2 at 6 and 9 h after spore
application. It was concluded that the negative
binomial distribution was found to be the most
acceptable model to fit the observed data sets
considering that P. penetrans spores are clumped.
This issue of over-dispersion with zeros exists in a
dataset [2] we recently analyzed. Based on this class
of distributions, we tested two approaches to adjust
the over-dispersed count data with zeros [3-5]. The
first approach was to scale the variance of the
Poisson distribution by submitting a dispersion
parameter and multiplying it by the variance. The
second approach was to test another probability
distribution to handle the count data dispersion, such
as the Negative binomial the Zero-inflated Poisson
(ZIP) or the Zero-inflated negative binomial (Zinb)
model.
Overall, in this paper, we employed and compare
these different models with a particular focus, on the
over-dispersed count data with zeros.
Moreover, this paper attempts to encourage
researchers dealing with biological data not to ignore
the over-dispersion which statistically influence the
conclusions by underestimating the variability of the
data.
2 Materials and Methods
2.1 Meloidogyne spp. Culture
A culture of M. javanica was maintained on tomato
plants (cherry tomato variety Tiny Tim) in the
glasshouse. Eggs were collected by dissolving the
gelatinous matrix into a solution of 0.5% sodium
hypochlorite (NaOCl) (10% commercial bleach),
passing the solution through a 200-mesh (75 µm)
sieve, nested over a 500-mesh (26 µm) sieve and
rinsing the eggs under slow running tap water to
remove residual NaOCl [6]. Second stage juveniles
(J2) were then hatched using standard laboratory
practices [7].
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2.2 P. penetrans Spore Attachment Data
Attachment tests on freshly hatched J2 were
conducted in 2.5-cm Petri dishes using standard
techniques [8]. Spore suspension of P. penetrans
(Nematech Co. Ltd Japan) were prepared in tap
water [9], and fresh J2s of root-knot nematodes were
exposed to 5000 spores per Petri dish [10]. All
dishes were placed in a 28°C incubator. Nematodes
were observed under an inverted microscope at ×200
magnification [10] and numbers of P. penetrans
spores attached per nematode were recorded (Table
1). For the data-set (Table 1), a total of 36 random
nematodes were examined for P. penetrans spore
attachment (att) after incubation of the Petri dishes
at 28°C for 1, 3, 6 and 9 h.
2.3 Statistical Analysis
In this research, we use a dataset Table 1 based on
the attachment counts of the bacterium P. penetrans
spores to root-knot nematodes cuticle. This dataset
which is presented in the results as Table 1, contains
four independent variables (time 1, 3, 6, and 9hrs)
and is analyzed with STATA 9.1 for Windows
Statistical Software [11], to demonstrate the
application of Poisson, negative binomial, over-
dispersed Poisson, and Zero-inflated Poisson
approach to modeling over-dispersion in count data
and explain with those models the excess zeroes [12-
14].
Moreover, to estimate these calculates the
commands used to produce the STATA output were
sum () for descriptive statistics, glm (),
family(poisson) nolog for poisson model, nbreg (),
nolog for Negative binomial regression and zip (),
inflate () nolog for zero-inflated Poisson model. All
the above commands are on Stata's manual, e.g.,
stata.com/help.cgi?poisson for the Poisson
regression, stata.com/help.cgi?nbreg for the
Negative binomial regression, stata.com/help.cgi?zip
for the Zero-inflated Poisson and
stata.com/help.cgi?zinb for the Zero-inflated
negative binomial.
3 Results
3.1 P. penetrans spore Attachment Data
The Table 1 shows the number P. penetrans spores
attached (att) to nematode cuticle. These data
characterized by excess zeros were used to model P.
penetrans’s attachment.
Table 1. Table showing the observed values of P.
penetrans spores attachment
Variable (h)
observed values attachment
(att)
1
0; 1; 3; 4; 3; 4; 1; 0; 2; 1; 1; 3; 3;
2; 1; 2; 2; 1; 1; 2; 0; 3; 2; 2; 1; 1;
1; 2; 0; 1; 3; 0; 3; 1; 1; 1
3
2; 0; 3; 4; 1; 7; 3; 1; 7; 6; 2; 4; 1;
7; 4; 1; 8; 5; 0; 8; 3; 6; 4; 7; 6; 2;
0; 4; 6; 4; 3; 8; 1; 0; 1; 5
6
1; 6; 6; 8; 0; 5; 15; 4; 0; 4; 5; 9; 6;
7; 7; 14; 7; 1; 9; 9; 0; 6; 8; 3; 5; 5;
4; 8; 10; 0; 0; 3; 7; 12; 4; 2
9
7; 2; 3; 9; 7; 3; 5; 6; 14; 12; 6; 10;
12; 2; 11; 3; 8; 19; 11; 9; 7; 6; 12;
8; 0; 0; 4; 13; 16; 3; 1; 8; 12; 3; 0;
10
3.2 Statistical Analysis of P. penetrans’s
Attachment
Descriptive statistics for counts variables (var1-4)
time 1, 3, 6 and 9hrs (h), were presented in Table 2.
The Table 2 shows that for var. 3h the mean is twice
the variance, and for var. 6h, and 9h the mean is
thrice the variance.
Table 2. Descriptive statistics for var1, var2, var3,
var4
Variable
(h)
Mean
Variance
Std.
Dev.
Min
Max
1
1.638
1.265
1.125
0
4
3
3.722
6.663
2.581
0
8
6
5.555
14.768
3.842
0
15
9
7.278
22.778
4.773
0
19
This suggests over-dispersion means the
assumptions of the Poisson model are not met and
make the Negative Binomial distribution a useful
over-dispersed alternative to the Poisson
distribution.
Based on the above observations we presume that
the variance is proportional rather than equal to the
mean, therefore I divide Pearson's chi-squared by its
d.f. in order to estimate the scale parameter φ.
Specifically, var(Y) = E(Y) = φµ applying that if φ
= 1, the variance equals the mean and the Poisson
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model is obtained. If the φ value is more than one (φ
> 1), the data are over-dispersed when compared to
Poisson.
The expected value of this statistic is n- p if the
model is correct. Equating the statistic to its
anticipation and solving for φ gives the estimate 1.
φˆ = χ2p / n− p estimate 1 (equation 1)
In our model Table 3 Generalized linear models,
χ2p = 292.272 for a model with p = 3 parameters on n
= 144 observations, which leads to φˆ = 292.272
/140 = 2.087. From Table 3 we have that the Poisson
estimates the standard errors by √2.087 = 1.4448,
which inflates them by 49.4%. The high value of
Pearson's χ2 and the p χ2 indicates a lack of fit,
which is not related to specification problems but
rather to over-dispersion. From Table 3, the
Pearson's χ2 statistic divided by its degree of
freedom (df) leads to 2.087 indicating over-
dispersion and however the Deviance statistic
divided by its df leads to 2.495 recommended over-
dispersion.
As shown above another method for modeling
over-dispersion in count data is to start with a
Poisson regression model and add a multiplicative
random effect θ to represent unobserved
heterogeneity,
This directs to the Negative Binomial regression
model (nbreg) as presented in Table 4. The Negative
Binomial model (nbreg), Table 4, gives estimates
that are very akin to the Poisson model, and retain
the same interpretation with average unobserved
characteristics as the count variables contain zeroes
Table 1.
The conditional probability distribution of the
outcome Y (equation 1), specified an unobserved θ
variable of Poisson with mean and variance θµ
(equation 1), proposed that the data of Table1 would
be Poisson if only we could observe θ. In our results,
we do the hypothesis that θ captures unobserved
factors that increase (if θ > 1) or decrease (if θ < 1).
In our results (Table 5), the output uses alpha which
is equal to 0.32245 to label the variance of the
unobservable, recommending that the data would not
be Poisson.
As was noted above in the STATA log the
Negative Binomial model provides estimates that are
very similar to the Poisson model Table 4, and have
the same interpretation. The standard errors
resemble the over-dispersed Poisson standard errors,
and both are more considerable than the reference
Poisson errors.
Maybe our distributional assumption is similar
but the nbreg STATA test of Table 5, leads to χ2LR =
69.04 which is highly significant (p=0.000)
suggested that the Negative Binomial is better than
the Poisson model. Furthermore, it is obvious that
AICPoisson, BICPoisson Table 3, and AICNeg.Binomial,
BICNeg.Binomial Table 6, has very different values
suggested the Negative Binomial model fits
considerably better than the Poisson, but even has
deviance suggested that empirical data probably
show fewer zeroes than would be expected under
either model. To solve that, we used a Zero-Inflated
(ZIP) Model for frequent zero-valued observations
and a zero-inflated negative binomial (ZINB) model
for modeling over-dispersion and excess zeros.
In an analysis of ZIP Table 7, the Poisson model
predicts only 8% of the 11,8% of the nematodes
without P. penetrans spores. The Poisson model
prognosticates 8%, so it underestimates the zeroes
by five percentage points.
The ZIP model proposes that there are two latent
categories of nematodes, the “always zero” and
another the “not always zero”, where the count has a
Poisson distribution with mean and variance > 0. To
solve it we were used in STATA the zero-inflated
Poisson command in the inflate() option Table 8.
The inflate equation of the outcome values Table 8,
shows that the related to the ZIP model where the
probability of “always zero” exists in 3, 6 and 9 hrs
of incubation. Table 8 shows that the incubation
period (3, 6 and 9hrs) is a significant predictor of
being in the “always zero” class. Furthermore, we
can observe from Table 9 that as alpha = 0 is
significantly different from zero signifying that our
data are over-dispersed and that a zero-inflated
negative binomial model is more proper than a zero-
inflated Poisson model. Moreover, the Zinb model in
the option zfitnb Table 10, solve the problem with
the excess zeroes, predicting that 10.5% of the tested
nematodes were unencumbered with P. penetrans
spores a very close to the observed (zobs) value of
11.8 Table 7.
Table 3. Poisson regression and a GLMs Poisson to
accommodate the excess variation.
STATA command poisson att i.h, vce(robust
Poisson regression
Number of obs =144
Wald chi2(3) =101.43
Prob > chi2 =0.0000
Log pseudolikelihood = -380.23145
Pseudo R2 =0.1667
att
Coef.
Robust
Std.
Err.
z
P>|z|
[95%
Conf.
Interval
]
h
3
.820
.160
5.10
0.000
.504
1.135
6
1.22
.160
7.60
0.000
.905
1.535
9
1.49
.156
9.52
0.000
1.183
1.797
_cons
.494
.113
4.36
0.000
.2721
.715
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STATA command glm att i.h, family(poisson) nolog
Generalized linear models
No. of obs = 144
Optimization
: ML
Residual df = 140
Deviance
= 349.353
(1/df) Deviance = 2.495
Pearson
= 292.272
(1/df) Pearson = 2.087
Variance function: V(u) = u
Link function : g(u) = ln(u)
[Poisson]
[Log]
Log likelihood
= -380.231
AIC = 5.336
BIC =-346.41
att
Coef.
OIM
Std.Err.
z
P>|z|
[95%
Conf.
Interval]
h
3
.820
.156
5.25
0.000
.514
1.126
6
1.22
.148
8.24
0.000
.930
1.511
9
1.49
.144
10.35
0.000
1.20
1.773
_cons
.494
.130
3.79
0.000
.238
.749
Table 4. Comparing estimates and standard errors
based on the Negative Binomial model and on the
Poisson regression model.
Variable
poisson
overdisp
nbreg
att
h
3
6
9
_cons
.82030
.82030
.82030
.15624
.22575
.20586
1.2207
1.2207
1.2207
.14815
.21406
.19979
1.4908
1.4908
1.4908
.14410
.20821
.1968
.49401
.49401
.49401
.13018
.1881
.16104
lnalpha _cons
-1.1286952
.22315033
Table 5. Negative Binomial regression model.
STATA command nbreg att i.h
Negative binomial regression
Number of obs = 144
LR chi2(3) = 53.91
Dispersion = mean
Prob > chi2 = 0.0000
Log likelihood = -345.70896
Pseudo R2 = 0.0723
att
Coef.
Std.
Err.
z
P>|z|
[95%
Conf.
Interv
al
h
3
.8203
.2058
3.98
0.000
.41680
1.2237
6
1.2207
.1997
6.11
0.000
.82918
1.6123
9
1.4908
.1968
7.57
0.000
1.1050
1.8765
_cons
.49401
.1610
3.07
0.002
.17838
.80965
/lnalpha
-1.128
.2231
-1.566
-
.69132
alpha
.32345
.0721
.20886
.50091
Likelihood-ratio test of alpha=0: chibar2(01) = 69.04
Prob>=chibar2 =0.000
Table 6. A Negative Binomial model to
accommodate the excess variation.
STATA command glm att i.h, family(nb `v') nolog
Generalized linear models
No. of obs = 144
Optimization : ML
Residual df = 140
Scale parameter = 1
Deviance = 164.3046481
1/df) Deviance = 1.173605
Pearson = 112.0212999
(1/df) Pearson = .8001521
Variance function: V(u) =
u+(.3235)u^2
function : g(u) = ln(u)
[Neg. Binomial] Link
[Log]
Log likelihood =-345.7089557
AIC = 4.857069
BIC = -531.4692
Table 7. Zero-Inflated Poisson (ZIP) model.
STATA command sum zobs zfitp
Variable
Obs
Mean
Std.Dev.
Min
Max
zobs
144
0.11855
0.3238
0
1
zfitp
144
0.08057
0.0807
0.000
69
0.19419
Table 8. Zero-Inflated Poisson (ZIP) model.
STATAcommand zip att i.h, inflate(h) vce(robust)
Zero-inflated Poisson regression
Number of obs = 144
Nonzero obs = 127
Zero obs = 17
Inflation model = logit
Wald chi2(3)= 128.01
Log pseudolikelihood = -
350.0632
Prob > chi2 = 0.0000
att
Coef.
Robust
Std.
Err.
z
P>|z|
[95%
Conf.
Interval]
att
h
3
.87726
.1518
5.78
0.000
.57966
1.1748
6
1.3274
.14411
9.21
0.000
1.045
1.6099
9
1.5371
.14561
10.56
0.000
1.2517
1.8226
_c
ons
.53439
.1096
4.88
0.000
.31956
.74921
inflate
h
_cons
.07184
.09275
0.77
0.439
-.10994
.25363
-
2.6985
.6298
-4.28
0.000
-3.9330
-1.4641
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Table 9. Zero-Inflated Negative Binomial (Zinb)
model.
STATA command zinb att i.h, inflate(h) vuong zip
Zero-inflated negative binomial
regression
Number of obs = 144
Nonzero obs = 127
Zero obs = 17
Inflation model = logit
LR chi2(3)= 68.18
Log pseudolikelihood = -338.4195
Prob > chi2 = 0.0000
att
Coef.
Std.
Err.
z
P>|z|
[95%
Conf.
Interval]
att
h
3
.8618
.18628
4.63
0.000
.49678
1.227016
6
1.3262
.17982
7.38
0.000
.97380
1.678686
9
1.5489
.17491
8.86
0.000
1.2060
1.891729
_cons
.51906
.14880
3.49
0.000
.22741
.810722
inflate
h
_cons
.11522
.13260
0.87
0.385
-.14468
.3751219
-3.143
.94447
-3.33
0.001
-4.9946
-1.292392
/lnalpha
-1.8996
.33253
-5.71
0.000
-2.5514
-1.247904
alpha
.14961
.04975
.07797
.2871059
Likelihood-ratio test of alpha=0: chibar2(01) = 23.29
Pr>=chibar2 = 0.0000
Vuong test of zinb vs. standard negative binomial: z = 1.74
Pr>z = 0.0410
Table 10. Zero-Inflated Negative Binomial (Zinb)
model with zfitnb option.
Variable
Obs
Mean
Std.Dev.
Min
Max
zfitnb
144
0.1051
0.9739
0.237
0.268
Finally, to choose between the Negative
Binomial and Zero-inflated models the Vuong test,
Table 9, suggests that the Zero-inflated negative
binomial model is a significant better Pr>z = 0.0410,
over a standard negative binomial model.
4 Discussion
According to our data (Table 1), the procedure
discussed above confirmed that the Zero-inflated
negative binomial model is the more appropriate
model to estimate P. penetrans spore attachment
count data. The Negative binomial model is also the
preferred model as time of exposure increased (e.g.
6 or 9 h), confirming the original conclusion
reported by Vagelas [2, 15] that the data are over-
dispersed within a specific time period (e.g. 6). As
the variance is greater than the mean [16-21],
examination of the variability using the Negative
binomial was an acceptable model to describe P.
penetrans over-dispersion as an aggregating
organism, leading to the conclusion that the bacteria
are clumped and clustered under natural conditions.
Moreover, in this paper, we used the Poisson, the
Negative binomial and the Zero-inflated Poisson
model [22], to deal with count data especially, when
the data are over-dispersed and contain excessive
zero counts [23]. In addition to the Poisson model,
we used the Negative binomial and Zero-inflated
models to the count data of Table 1. The first step
should be a summary of Statistics (e.g. the observed
data's sample mean and variance) to measure if the
data are over-dispersed. Second, we illustrated the
estimate of the dispersion parameter [24], as
deviance or Pearson's χ2 statistic divided by the
degrees of freedom, which is often used to indicate
over-dispersion for Poisson models. Third, as the
Poisson model is a special case of the Negative
Binomial when σ2 = 0, we apply a likelihood ratio
test to compare the two models. Finally, we
suggested that the Zero-inflated models should be
applied to estimate the dispersion parameters of the
Negative Binomial [25] and offer an explanation for
the excessive zero counts [26-27].
5 Conclusion
In biological research including biocontrol, count
data with a large proportion of zeros are often
recorded [28-30]. For those count data Poisson
regression is often used when the conditional
variance is equal to the mean (equi-dispersion), or
alternative since the variance exceeds the mean
(over-dispersion), the negative binomial model was
used to fit the count data. Since the over-dispersion
has occurred, the probability of a structural zero,
true zeros, and excess zero needs to test further.
Further, apart from the negative binomial model, the
Zero-inflated models can be an alternative to
overcome over-dispersed data and can be used as a
model to offer an explanation for the excess zeros
condition in the counts.
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